Mirror Confinement

Mirror confinement (magnetic mirror) is a form of nuclear fusion device that uses a linear magnetic field configuration to confine plasma. The name derives from the use of the “magnetic mirror effect,” which reflects charged particles through magnetic field gradients. Compared to toroidal devices, it has the inherent advantages of simpler structure and the theoretical possibility of steady-state operation and direct energy conversion.

The concept of mirror confinement was proposed in the dawn of fusion research in the 1950s and developed through decades of theoretical and experimental research. In the 1970s, the invention of the tandem mirror dramatically improved confinement performance, and it was once considered a leading candidate for fusion reactors alongside the tokamak. However, following program reductions in the 1980s, research continues today at only a limited number of facilities. Recently, advances in high-temperature superconducting technology and the entry of private companies have brought renewed attention to the mirror approach.

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Principles of Mirror Confinement

The Magnetic Bottle Concept

The essence of mirror confinement lies in forming a “magnetic bottle” that uses magnetic field gradients to confine charged particles to a specific spatial region. When two solenoid coils are arranged at an appropriate spacing, the magnetic field is strong near the coils and weak in the central region between them. This magnetic field geometry causes charged particles entering the weak-field central region to be reflected at the strong-field regions at both ends, creating a state as if confined within a bottle.

The concept of the magnetic bottle was first proposed in 1952 by Enrico Fermi in his research on the origin of cosmic rays. Fermi considered a mechanism whereby charged particles are reflected and accelerated by magnetic field clouds (magnetic bottles) in intergalactic space. This concept of “Fermi acceleration” became the theoretical foundation for mirror confinement in subsequent fusion research.

Earth’s magnetosphere is also an example of a giant magnetic bottle in nature. Due to the structure where magnetic field lines converge near the north and south poles (magnetic mirror points), charged particles arriving from space are captured in the Van Allen belts around Earth. The aurora phenomenon occurs when some of these charged particles collide with atmospheric molecules in the polar regions and emit light.

Helical Motion of Charged Particles

To understand the motion of charged particles in a magnetic field, we must first consider the action of the Lorentz force. When a particle with charge $q$ and mass $m$ moves with velocity $\mathbf{v}$ in a magnetic field with magnetic flux density $\mathbf{B}$, the force acting on the particle is given by:

$$\mathbf{F} = q\mathbf{v} \times \mathbf{B}$$

Since this force is always perpendicular to the velocity vector, the particle’s energy is conserved. In a uniform magnetic field, particles spiral around the magnetic field lines. The angular frequency of this gyration (cyclotron angular frequency or Larmor angular frequency) is expressed as:

$$\omega_c = \frac{|q|B}{m}$$

For deuterium ions, the cyclotron frequency in a 1 Tesla magnetic field is approximately 7.6 MHz. Electrons are about 1836 times lighter than ions, so they gyrate at a much higher frequency in the same magnetic field.

The radius of gyration (Larmor radius or cyclotron radius) is expressed using the velocity component perpendicular to the magnetic field line $v_\perp$ as:

$$r_L = \frac{mv_\perp}{|q|B}$$

In a typical fusion plasma (ion temperature 10 keV, magnetic field 3 T), the Larmor radius of deuterium ions is approximately 4 mm. This scale is sufficiently small compared to the device size (several meters), so particles can be approximated as moving along the magnetic field lines.

Magnetic Moment and Adiabatic Invariant

A spiraling charged particle forms a small current loop through its gyration. The magnetic moment $\mu$ associated with this current loop is defined as:

$$\mu = \frac{mv_\perp^2}{2B} = \frac{W_\perp}{B}$$

where $W_\perp = mv_\perp^2/2$ is the energy of motion perpendicular to the magnetic field line.

The magnetic moment has the property of being an adiabatic invariant. That is, when the magnetic field changes slowly compared to the cyclotron period, the particle’s magnetic moment is conserved:

$$\mu = \frac{mv_\perp^2}{2B} = \text{const.}$$

The condition for this adiabatic invariance requires that the relative change in magnetic field during one gyration be small:

$$\frac{1}{B}\left|\frac{dB}{dt}\right| \ll \omega_c$$

Or for spatial variation:

$$\frac{r_L}{B}\left|\frac{\partial B}{\partial s}\right| \ll 1$$

where $s$ is the distance along the magnetic field line. In fusion plasmas, this condition is usually well satisfied, allowing analysis based on the conservation of magnetic moment.

Conservation of magnetic moment can be understood as a special case of the adiabatic theorem. It is the Hamiltonian mechanics version of the adiabatic theorem in quantum mechanics, corresponding to the invariance of the action integral $J = \oint p \, dq$ for slow changes in system parameters. For charged particle gyration, this action integral is proportional to the magnetic moment:

$$J_\perp = \oint p_\perp \, dl_\perp = 2\pi m v_\perp r_L = \frac{2\pi m^2 v_\perp^2}{|q|B} \propto \mu$$

Mechanism of Mirror Reflection

From the conservation of magnetic moment, we can derive the mechanism of mirror reflection. When a particle moves through a region where the magnetic field varies spatially, the magnetic moment $\mu$ is conserved, so as the magnetic field increases, the perpendicular velocity $v_\perp$ increases:

$$v_\perp^2 = \frac{2\mu B}{m}$$

Meanwhile, the particle’s total kinetic energy is conserved (ignoring collisions):

$$W = \frac{1}{2}m(v_\parallel^2 + v_\perp^2) = \text{const.}$$

From these relationships, the velocity component parallel to the magnetic field line $v_\parallel$ is:

$$v_\parallel^2 = \frac{2W}{m} - v_\perp^2 = \frac{2W}{m} - \frac{2\mu B}{m} = \frac{2(W - \mu B)}{m}$$

At the point where the magnetic field becomes strong enough that $\mu B = W$, $v_\parallel = 0$. This is the mirror point (reflection point). Beyond the mirror point, $v_\parallel^2 < 0$ becomes physically impossible, so the particle is reflected at this point.

The force acting on the particle during reflection is given by the following expression based on magnetic moment:

$$F_\parallel = -\mu \frac{\partial B}{\partial s} = -\mu \nabla_\parallel B$$

This force is proportional to the magnetic field gradient and always acts in the direction toward the weak-field region. In the direction of increasing magnetic field ($\nabla_\parallel B > 0$), it decelerates the particle and eventually reflects it.

Mirror Ratio and Confinement Conditions

An important parameter for mirror magnetic fields is the mirror ratio $R_m$, defined as:

$$R_m = \frac{B_{\max}}{B_{\min}}$$

where $B_{\max}$ is the maximum magnetic field at the ends (mirror points) and $B_{\min}$ is the minimum magnetic field in the central region.

We derive the condition for a particle in the weak-field region (center) to be reflected at the mirror point. Let the velocity of a particle at the center (magnetic field $B_0$) be $(v_{\parallel 0}, v_{\perp 0})$, with total velocity $v_0 = \sqrt{v_{\parallel 0}^2 + v_{\perp 0}^2}$. The pitch angle $\theta$ is defined as:

$$\sin\theta = \frac{v_{\perp 0}}{v_0}$$

From the conservation of magnetic moment and energy, the condition for reflection at the mirror point (magnetic field $B_m$) is:

$$v_{\parallel m}^2 = v_0^2 - \frac{B_m}{B_0}v_{\perp 0}^2 = 0$$

From this:

$$\sin^2\theta = \frac{v_{\perp 0}^2}{v_0^2} = \frac{B_0}{B_m} = \frac{1}{R_m}$$

Therefore, the critical pitch angle $\theta_c$ is:

$$\theta_c = \arcsin\left(\frac{1}{\sqrt{R_m}}\right)$$

Particles with $\theta > \theta_c$ (i.e., $v_\perp / v > 1/\sqrt{R_m}$) are reflected and confined. On the other hand, particles with $\theta < \theta_c$ pass through the mirror point without reflection and escape from the ends.

In typical mirror devices, the mirror ratio is $R_m = 2 \sim 5$, and the critical angle is $\theta_c = 45° \sim 27°$.

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Particle Orbits and Loss Cone

Representation in Velocity Space

To visualize particle confinement conditions, representation in velocity space is useful. Considering three-dimensional velocity space $(v_x, v_y, v_z)$ with the magnetic field line direction as the $z$-axis, axial symmetry allows representation in the two-dimensional plane $(v_\perp, v_\parallel)$, where:

$$v_\perp = \sqrt{v_x^2 + v_y^2}, \quad v_\parallel = v_z$$

In this velocity space, we consider the regions of confined and escaping particles. Particles with constant total velocity $v$ are distributed on a sphere of radius $v$ (a circle in the two-dimensional representation) in velocity space.

Expressing the confinement condition $\sin^2\theta > 1/R_m$ in velocity space:

$$\frac{v_\perp^2}{v^2} > \frac{1}{R_m}$$

This becomes:

$$v_\perp^2 > \frac{v_\parallel^2 + v_\perp^2}{R_m}$$

Rearranging:

$$v_\perp^2 > \frac{v_\parallel^2}{R_m - 1}$$

Or:

$$\frac{v_\parallel^2}{v_\perp^2} < R_m - 1$$

Geometry of the Loss Cone

The region not satisfying the above inequality, i.e.:

$$\frac{v_\parallel^2}{v_\perp^2} \geq R_m - 1$$

represents particles that are not confined. Expressing this region in velocity space yields a double cone with the $v_\parallel$ axis (magnetic field line direction) as its axis. This cone is called the “loss cone.”

The half-apex angle $\alpha$ of the loss cone equals the complement of the critical pitch angle $\theta_c$:

$$\alpha = 90° - \theta_c = \arccos\left(\frac{1}{\sqrt{R_m}}\right)$$

Or:

$$\tan\alpha = \sqrt{R_m - 1}$$

For mirror ratio $R_m = 2$, $\alpha = 45°$, meaning approximately half of velocity space becomes the loss cone region. For $R_m = 10$, $\alpha \approx 72°$ and the loss cone narrows, but still causes significant losses.

In a simple mirror configuration with mirrors at both ends, loss cones exist for both positive and negative $v_\parallel$ directions, forming an overall double-cone loss region.

Loss Cone Filling by Collisions

In a collisionless plasma, particles initially distributed outside the loss cone are confined indefinitely. However, in real plasmas, Coulomb collisions continuously cause small-angle scattering of particle velocity vectors. Through this process, particles outside the loss cone gradually scatter into the loss cone and escape from the ends.

The 90-degree deflection time $\tau_{90}$ due to Coulomb collisions is approximated by:

$$\tau_{90} = \frac{12\pi^{3/2}\varepsilon_0^2 m^{1/2} T^{3/2}}{n q^4 \ln\Lambda}$$

where $T$ is the temperature (in energy units), $n$ is the density, and $\ln\Lambda \approx 15 \sim 20$ is the Coulomb logarithm.

For deuterium ions (temperature 10 keV, density $10^{20}$ m$^{-3}$), $\tau_{90} \approx 1$ second. Since the loss cone occupies a finite solid angle in velocity space, the typical time scale for particles to scatter into the loss cone and be lost (confinement time) is:

$$\tau_p \sim \tau_{90} \ln R_m$$

The logarithmic factor $\ln R_m$ arises because when the loss cone solid angle is small, many small-angle scatterings are required for a particle to reach the loss cone boundary.

Confinement Time Scaling

Theoretical evaluation of particle confinement time in simple mirrors was performed in detail by Pastukhov. For high-temperature, low-collisionality plasmas (where the mean free path is longer than the mirror length), the confinement time scales as:

$$\tau_p \approx \tau_{ii} \cdot R_m \cdot \ln R_m \cdot g(\phi_c/T_i)$$

where $\tau_{ii}$ is the ion-ion collision time and $g$ is a function depending on the plug potential $\phi_c$. For a simple mirror without potential confinement ($\phi_c = 0$), $g \sim 1$ and:

$$\tau_p \approx \tau_{ii} \cdot R_m \cdot \ln R_m$$

This equation shows that mirror confinement time is inherently limited by collision time. To achieve fusion conditions ($n\tau > 10^{20}$ m$^{-3}$s), very high temperatures (low collision frequency) would be required in a simple mirror.

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MHD Stability

Flute Instability

In simple mirror configurations, plasma suffers from serious magnetohydrodynamic (MHD) instabilities. The most important of these is the “flute instability.” The name derives from the shape of the unstable mode, which resembles the fluting pattern carved into Greek architectural columns.

Flute instability is driven by the interaction between magnetic field line curvature and pressure gradients. When magnetic field lines have outwardly convex curvature (the center of curvature is outside the plasma) and pressure decreases radially outward, the plasma becomes unstable.

The physical mechanism of this instability can be understood as follows. When small irregularities form on the plasma surface, the magnetic field line length increases at convex portions and the magnetic field strength decreases. At constant pressure, the plasma tends to expand more easily at convex portions, causing the perturbation to grow. This process proceeds in a cascade, and the plasma rapidly escapes across field lines to the outside.

The growth rate $\gamma$ of flute instability typically has a scale determined by the Alfven velocity and curvature radius:

$$\gamma \sim \sqrt{\frac{2p}{R_c \rho}}$$

where $p$ is plasma pressure, $R_c$ is the radius of curvature of the magnetic field lines, and $\rho$ is mass density.

Stability Conditions and Minimum-B Configuration

A sufficient condition for MHD stability is the “minimum-B configuration.” This is a configuration where the magnetic field is minimum at the center of the confinement region and increases outward in all directions.

The minimum-B condition is expressed as:

$$\nabla_\perp |B|^2 > 0 \quad \text{(in all directions within the plasma region)}$$

When this condition is satisfied, the plasma is stably held at the bottom of a magnetic field “well.”

In a simple mirror (axisymmetric solenoid configuration), the magnetic field increases axially, but there are regions where the magnetic field decreases with distance from the center radially. This is the cause of flute instability.

Baseball Coils and Yin-Yang Coils

As innovative coil designs to achieve minimum-B configuration, “baseball coils” and “Yin-Yang coils” were developed in the 1960s.

Baseball coils are single coils shaped like the seams of a baseball. These coils have a three-dimensionally twisted shape, and the generated magnetic field breaks axial symmetry. At the center of the baseball coil, the magnetic field is minimum and increases outward in all directions, achieving a minimum-B configuration.

Yin-Yang coils consist of two C-shaped coils arranged perpendicular to each other. The name derives from the shape resembling the Chinese Taiji symbol, with the two coils interlocking. Like baseball coils, Yin-Yang coils form a minimum-B configuration and have the advantage of achieving higher mirror ratios.

The magnetic pressure surfaces generated by these coils have complex shapes where elliptical cross-sections rotate 90 degrees along the axis, unlike the cylindrical shape of axisymmetric solenoids. This non-axisymmetry is essential for stabilization.

The Anchor Concept

In tandem mirrors, special regions called “anchors” are placed at both ends of the central cell to ensure MHD stability. Anchors are mirror cells with minimum-B configuration, typically constructed with baseball coils or Yin-Yang coils.

The growth of MHD instability is determined by the pressure-weighted average curvature along the magnetic field line:

$$\int \kappa \cdot \nabla p \, dl > 0 \quad \Rightarrow \quad \text{unstable}$$

where $\kappa$ is the curvature vector of the magnetic field line, $p$ is plasma pressure, and the integration is performed along the field line.

In anchor regions, the “good curvature” (center of curvature on the plasma side) contributes negatively to this integral. If the negative contribution from anchors can offset the positive contribution from “bad curvature” regions in the central cell, the overall system becomes stable.

For this stabilization mechanism to work effectively, the plasma pressure in the anchor region must be maintained sufficiently high. In tandem mirror devices, electron cyclotron resonance heating (ECH) or neutral beam injection (NBI) is used to heat the plasma in the anchor regions.

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End Losses and Suppression Techniques

Physics of End Losses

The fundamental challenge of the mirror approach is particle and energy loss from the open ends (end losses). Particles in the loss cone pass through the mirrors at both ends and escape, limiting plasma confinement.

The particle flux due to end losses $\Gamma_{\text{end}}$ is approximately evaluated as:

$$\Gamma_{\text{end}} \sim \frac{n}{\tau_p} \sim \frac{n}{\tau_{ii} R_m \ln R_m}$$

To compensate for these losses, continuous fuel supply through neutral beam injection or pellet injection is required.

Regarding energy losses, in addition to energy carried away by particles escaping from the ends, axial thermal conduction by electrons is also important. Hot electrons transport heat along the magnetic field lines to the ends, limiting energy confinement.

The confinement parameter (nτE product) of simple mirrors is typically around $10^{18}$ m$^{-3}$s, more than two orders of magnitude below fusion conditions (>$10^{20}$ m$^{-3}$s for D-T reactions). This inherent limitation motivated the development of more advanced confinement concepts.

Electrostatic Confinement

An innovative approach to suppress end losses, electrostatic potential confinement, was proposed in the 1970s.

In plasma, electrons have much higher thermal velocities than ions. In the collisionless case, electrons tend to escape from the ends faster than ions, but this creates a positive space charge that pulls electrons back, forming a self-consistent potential.

However, in simple mirrors, this spontaneous potential (sheath potential) is only on the order of the electron temperature ($\phi \sim T_e/e$), which is insufficient to improve ion confinement.

In tandem mirrors, a high positive potential is intentionally formed in “plug cells” at both ends to electrostatically confine ions. When the plug potential $\phi_c$ is sufficiently larger than the ion temperature $T_i$, ions must overcome the potential barrier to escape from the ends, dramatically improving confinement time:

$$\tau_p \sim \tau_{ii} \cdot R_m \cdot \exp\left(\frac{e\phi_c}{T_i}\right)$$

Due to the exponential dependence, even $\phi_c \sim 3 T_i$ improves confinement time by more than 20 times.

To form the plug potential, methods that selectively heat electrons in the plug cell are used. By generating hot electrons through electron cyclotron heating (ECH), some of them escape from the mirror ends, leaving behind positive charges and forming a positive potential.

Thermal Barrier

A concept to further improve the efficiency of potential confinement, the “thermal barrier,” was proposed in 1979 by Baldwin and Logan at Lawrence Livermore National Laboratory.

Normally, when plug cells and central cells are electrically connected, electrons in both regions tend to mix toward thermal equilibrium. This causes hot electrons heated in the plug cell to flow into the central cell, increasing the heating power required to maintain the plug potential.

The thermal barrier forms a local potential trough (negative potential) between the plug cell and central cell, suppressing electron mixing. This potential trough functions as a “thermal barrier” that prevents electrons from flowing from the central cell to the plug cell.

To form a thermal barrier, electron density must be reduced in the barrier region. This is achieved by ion pumping with neutral beams (removing ions through charge exchange reactions) or by selectively heating electrons with ECH to promote their outflow.

With thermal barriers, tandem mirrors can maintain high plug potentials while significantly reducing required heating power. This concept was incorporated into the designs of major tandem mirror experiments of the 1980s (TMX-U, MFTF-B, GAMMA 10).

Stabilization and Confinement through Rotation

Confinement improvement using plasma rotation has also been studied. In axisymmetric mirror configurations, when plasma is given azimuthal rotation, centrifugal force forms an additional potential well that confines ions.

The change in potential energy due to rotation velocity $v_\theta$ is:

$$U_{\text{cent}} = -\frac{1}{2}m_i v_\theta^2 = -\frac{1}{2}m_i \omega^2 r^2$$

When the rotation velocity is comparable to the ion thermal velocity ($v_\theta \sim v_{ti}$), the potential due to centrifugal force becomes comparable to the ion temperature, and significant confinement improvement can be expected.

However, rotating plasmas are accompanied by inherent instabilities such as centrifugal instability, so there is an optimal rotation velocity. Also, torque input to maintain rotation (electrode biasing, oblique beam injection, etc.) is required.

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Tandem Mirror

Concept Proposal and Development

The tandem mirror concept was proposed in 1976 independently by Dimov, Zakaidakov, and Kishinevsky in the Soviet Union and by Fowler and Logan in the United States. This was a groundbreaking idea to solve the end loss problem of simple mirrors.

The basic idea is to connect multiple mirror cells in a linear arrangement (tandem configuration) and form potential barriers at both ends to confine ions. The main plasma is confined in the central “central cell,” and high potentials are generated in the “plug cells” at both ends.

In early tandem mirror concepts, the potential difference was generated by confining plasma with higher density and temperature than the central cell in the plug cells. Later, with the introduction of the thermal barrier concept, more efficient potential confinement became possible.

Components of Tandem Mirrors

A typical tandem mirror device consists of the following functional cells:

Central Cell The central region that confines the main plasma. It is composed of an axisymmetric solenoid magnetic field and forms the longest plasma column. Fusion reactions occur in this region. The magnetic field strength is about 0.5-2 T, and the length ranges from 10 to 100 m depending on the device.

Plug Cell Located at both ends of the central cell, this region forms the positive plug potential. Electrons are locally heated by electron cyclotron heating (ECH), and the potential is generated by accumulation of positive charge through electron outflow. Strong-field coils are placed to increase the mirror ratio.

Barrier Cell In designs with thermal barriers, this is placed between the plug cell and central cell. It lowers electron density to form a potential trough and achieves thermal insulation of electrons.

Anchor Cell A minimum-B region for MHD stabilization. It is composed of baseball coils or Yin-Yang coils. It may be incorporated into the plug cell or placed as an independent cell.

End Cell Located at the very ends of the device, this region handles escaping plasma and performs direct energy conversion. In future power plants, direct converters that convert the energy of escaping particles directly to electricity are planned for installation.

Confinement Scaling Laws

The ion confinement time in tandem mirrors strongly depends on the ratio of plug potential $\phi_c$ to ion temperature $T_i$:

$$\tau_i \propto \tau_{ii} \cdot R_m \cdot \left(\frac{e\phi_c}{T_i}\right) \cdot \exp\left(\frac{e\phi_c}{T_i}\right)$$

The plug potential is proportional to the difference between the electron temperature in the plug cell $T_{ep}$ and the electron temperature in the central cell $T_{ec}$:

$$\phi_c \approx (T_{ep} - T_{ec})/e \cdot f(\text{density ratio})$$

Therefore, maintaining high electron temperature in the plug cell is important for confinement improvement.

The energy confinement time $\tau_E$ must account for losses due to axial thermal conduction by electrons in addition to ion end losses. The loss power due to electron thermal conduction is:

$$P_e \sim n T_e \cdot \frac{L}{\tau_{ee}} \cdot \frac{1}{R_m}$$

where $L$ is the length of the central cell and $\tau_{ee}$ is the electron-electron collision time.

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Historical Development

1950s: Birth of the Mirror Confinement Concept

In the 1950s, the dawn of fusion research, mirror confinement was considered one of the most promising approaches.

In 1952, Richard F. Post at Lawrence Livermore National Laboratory (then the Livermore branch of the University of California Radiation Laboratory) began research on plasma confinement using magnetic mirrors. Post was inspired by the capture of charged particles by Earth’s magnetic field (Van Allen belts) and proposed confining fusion plasma in an artificial magnetic bottle.

At the same time, G.I. Budker began research on mirror confinement at the Institute of Nuclear Physics in Novosibirsk in the Soviet Union. The independent development of the mirror concept in both the US and USSR demonstrates the physical reasonableness of this approach.

At the 2nd International Conference on the Peaceful Uses of Atomic Energy in Geneva in 1958, American and Soviet fusion research was publicly disclosed for the first time. Results on mirror confinement were also presented, marking the beginning of international research cooperation.

1960s: Establishment of Basic Physics

The 1960s was an era when the basic physics of mirror confinement was established.

At Lawrence Livermore, experimental devices such as Table Top, Toy Top, Alice, and 2X were built in succession. In these devices, basic physics such as conservation of magnetic moment, loss cone losses, and confinement limits due to collisions were experimentally verified.

In 1961, the theory of flute instability was established by M.N. Rosenbluth and others. It became clear that simple mirrors could not achieve fusion conditions due to this instability, and the importance of minimum-B configuration was recognized.

In 1963, the experimental device “Baseball” using baseball coils with minimum-B configuration was built at Livermore, and MHD-stable plasma confinement was demonstrated for the first time. This success determined the direction of mirror research.

In the late 1960s, high-temperature plasma was generated by neutral beam injection in the “2XIIB” device using Yin-Yang coils. Confinement of high-beta plasma with ion temperatures above 10 keV and beta values exceeding 50% was achieved.

1970s: Invention of the Tandem Mirror

The 1970s was an era when mirror research made great strides through the invention and demonstration of the tandem mirror concept.

In 1976, Dimov et al. in the Soviet Union and Fowler and Logan in the United States independently proposed the tandem mirror. This concept showed the possibility of overcoming the fundamental limitation of end losses in simple mirrors.

In 1979, the tandem mirror experimental device “TMX (Tandem Mirror Experiment)” was completed at Livermore. Although a relatively small device with a total length of 10 m and central cell length of 4 m, it succeeded in demonstrating the principle of potential confinement as the world’s first tandem mirror experiment.

In the same year, the thermal barrier concept was proposed by Baldwin and Logan. This showed that the confinement efficiency of tandem mirrors could be greatly improved theoretically.

In Japan, GAMMA 6 was completed at the University of Tsukuba in 1978, and tandem mirror research began.

1980s: Construction of Large Devices and Setbacks

The 1980s was an era of both the peak of tandem mirror research and major setbacks.

At Livermore, “TMX-U (TMX Upgrade),” an improved version of TMX, was completed in 1983. With this device having a thermal barrier configuration, simultaneous formation of potential confinement and thermal barrier was demonstrated.

An even more ambitious plan, the construction of “MFTF-B (Mirror Fusion Test Facility-B),” proceeded. With a total length of 54 m and magnetic field strength of 7 T, it was the world’s largest mirror device, aiming to achieve scientific breakeven (Q=1).

However, in 1986, although MFTF construction was completed, the program was canceled for budget reasons without ever being operated. The abandonment of a completed device became one of the greatest setbacks in fusion research history, stemming from energy research budget cuts under the Reagan administration and concentration of resources on tokamaks (especially TFTR, JET, JT-60).

This decision effectively ended mirror research in the United States. Many researchers moved to tokamaks and other fields, and the knowledge base of mirror confinement was severely damaged.

In the Soviet Union, research on the Gas Dynamic Trap (GDT) continued at Novosibirsk. GDT is an approach different from conventional mirrors, confining high-density, highly collisional plasma in an axisymmetric configuration.

In Japan, GAMMA 10 was completed at the University of Tsukuba in 1985. After the cancellation of TMX-U, it became the world’s largest tandem mirror and continues operation to this day.

1990s-2000s: Continued Research During Contraction

Even after the cancellation of major mirror programs, research continued at limited facilities.

At the University of Tsukuba’s GAMMA 10, detailed research on potential confinement proceeded. In 1998, Miyoshi et al. clearly demonstrated the formation of ion confining potential, confirming confinement improvement. This result validated the correctness of the tandem mirror concept.

At Russia’s GDT, high-beta plasma confinement in axisymmetric mirror configuration was studied. Heating and stabilization by oblique neutral beam injection succeeded, achieving stable confinement with beta values exceeding 60%.

At the University of Wisconsin, a GAMMA-10 scale device plan for basic research on mirror physics was considered.

2010s-Present: Signs of Revival

Recently, mirror confinement is attracting renewed attention.

Advances in high-temperature superconducting (HTS) technology have made magnetic field strengths above 20 T achievable, which were difficult with conventional low-temperature superconductors. Higher magnetic fields directly translate to improved mirror ratios and better confinement, increasing the competitiveness of the mirror approach.

The rise of private fusion companies is also an important factor. TAE Technologies is developing a unique concept combining field-reversed configuration (FRC) with mirror confinement.

At the University of Wisconsin, modern research on axisymmetric mirrors has resumed through the WHAM initiative. Conceptual design of advanced mirror devices using high-temperature superconducting magnets is underway.

At Russia’s GDT, applications of mirrors as neutron sources for fusion-fission hybrid reactors are being studied.

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Major Devices

2X / 2XIIB (USA, 1960s-1970s)

A series of simple mirror devices operated at Lawrence Livermore National Laboratory. 2X (completed 1965) began plasma heating by neutral beam injection, and 2XIIB (completed 1976) adopted a minimum-B configuration with Yin-Yang coils.

The following groundbreaking results were obtained at 2XIIB:

• Achievement of ion temperature of 13 keV
• High-beta plasma confinement with beta values over 50%
• Demonstration of the effectiveness of neutral beam heating
• Demonstration of MHD stabilization by minimum-B configuration

These results became the foundation for subsequent tandem mirror development.

TMX / TMX-U (USA, 1979-1987)

TMX (Tandem Mirror Experiment)

The world’s first tandem mirror device, completed in 1979 at Lawrence Livermore National Laboratory.

Parameter	Value
Total length	10 m
Central cell length	4 m
Central cell magnetic field	0.1 T
Plug magnetic field	1 T
Heating method	NBI

TMX experimentally demonstrated the principle of potential confinement for the first time. Potential formation in the plug cell and resulting improvement in ion confinement in the central cell were confirmed.

TMX-U (TMX Upgrade)

An improved version of TMX completed in 1983. It was designed incorporating the thermal barrier concept, aiming for more efficient potential confinement.

Parameter	Value
Total length	15 m
Central cell length	6 m
Mirror ratio	6
Heating methods	NBI, ECH

The following results were obtained at TMX-U:

• Formation and demonstration of thermal barrier
• Achievement of plug potential over 1 kV
• Improvement of confinement time by more than 10 times

Operation ended in 1987, and integration into the MFTF program was planned, but MFTF itself was canceled.

MFTF / MFTF-B (USA, 1980s)

MFTF (Mirror Fusion Test Facility)

The original plan was a simple mirror device using Yin-Yang coils, but following the success of the tandem mirror concept, the design was changed to “MFTF-B.”

MFTF-B

Parameter	Value
Total length	54 m
Central cell length	30 m
Central cell magnetic field	0.5 T
Mirror magnetic field (Yin-Yang coils)	7 T
Heating power	NBI 30 MW, ECH 6 MW
Target	Q = 1 (scientific breakeven)

MFTF-B was an ambitious project aiming for scientific breakeven (where input energy equals generated fusion energy).

In 1986, construction was completed at a cost of $372 million, but the program was canceled due to federal budget cuts without ever being operated. The abandonment of a completed device as-is became one of the greatest setbacks in fusion research history.

GAMMA 10 / GAMMA 10/PDX (Japan, 1985-Present)

The world’s largest tandem mirror device operated at the Plasma Research Center, University of Tsukuba.

Parameter	Value
Total length	27 m
Central cell length	6 m
Central cell magnetic field	0.4 T
Anchor magnetic field	0.6 T
Mirror magnetic field (maximum)	3 T
Heating methods	ICRF (ion cyclotron waves), ECH, NBI
Plasma density	$2 \times 10^{18}$ m$^{-3}$
Ion temperature	Over 10 keV (100 million degrees)
Electron temperature	1 keV (approximately 10 million degrees)

Major achievements of GAMMA 10:

Demonstration of Potential Confinement (1998-) Miyoshi et al. demonstrated clear formation of ion confining potential for the first time in the world. By heating the plug cell with ECH, they successfully formed a plug potential higher than the central cell and suppressed ion end losses.

Achievement of 100 Million Degree Ion Temperature Ion temperatures exceeding 10 keV were achieved by ion cyclotron resonance heating (ICRF). This recorded the highest temperature for a tandem mirror.

Confinement Improvement by Radial Electric Field Shear Since 2003, a phenomenon where radial transport of plasma is suppressed by spatial variation of the radial electric field (electric field shear) has been discovered. This is similar physics to H-mode and internal transport barriers (ITB) in tokamaks and has attracted attention as a universal phenomenon beyond device configurations.

Divertor Simulator Research (PDX) Since 2013, the device has been operated as a divertor plasma experimental device (GAMMA 10/PDX) utilizing the device ends. It contributes to basic research on divertor heat loads and plasma-wall interactions that are problematic in large tokamaks like ITER.

GDT (Russia, 1986-Present)

The Gas Dynamic Trap operated at the Budker Institute of Nuclear Physics in Novosibirsk.

Parameter	Value
Total length	7 m
Distance between mirrors	3 m
Central magnetic field	0.3 T
Mirror magnetic field	12 T
Mirror ratio	40
Heating method	NBI (oblique injection)
Beta value	Maximum 60%

GDT employs a different approach from conventional tandem mirrors. With high mirror ratio (R > 10) and high density (over $10^{19}$ m$^{-3}$), collisional plasma is confined. Under these conditions, the loss cone distribution rapidly relaxes to a Maxwellian distribution through collisions, and end losses are dominated by hydrodynamic outflow (gas dynamic confinement).

Characteristic achievements of GDT:

High-Beta Confinement in Axisymmetric Configuration By oblique neutral beam injection, ions are given a high perpendicular velocity component. This creates pressure anisotropy in the magnetic field line direction, suppressing flute instability even in axisymmetric configurations. Stable confinement with beta values exceeding 60% was achieved.

Application as Neutron Source GDT is being considered for application as a high-intensity neutron source, either as a driver for fusion-fission hybrid reactors or as a neutron source for materials testing.

C-2W / Norman (TAE Technologies, USA, 2017-Present)

A private fusion device operated by TAE Technologies (formerly Tri Alpha Energy).

Parameter	Value
Total length	24 m
Magnetic configuration	FRC (Field-Reversed Configuration) + Mirror confinement
Plasma temperature	50-80 MK
Confinement time	Over 30 ms

TAE employs a unique approach combining field-reversed configuration (FRC) with mirror confinement. FRC is a type of reversed field pinch, but with both ends closed by mirror magnetic fields.

Through neutral beam injection and external magnetic field control, FRC plasma lifetime and temperature have been dramatically improved. The ultimate goal is neutron-free fusion using proton-boron-11 (p-$^{11}$B) reactions.

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Modern Mirror Research

TAE Technologies

TAE Technologies (Foothill Ranch, California) is one of the most well-funded private fusion companies in the world. As of 2023, it has raised over $1.2 billion cumulatively.

TAE’s approach does not directly inherit conventional mirror confinement, but shares the commonality of having open-ended magnetic field configurations. In the main device “C-2W (nicknamed Norman),” FRC plasma is confined by mirror magnetic fields at both ends.

TAE’s proprietary technologies:

Beam-Driven FRC Neutral beam injection simultaneously heats and stabilizes FRC plasma. Conventional FRCs had short lifetimes (sub-millisecond), but TAE’s technology has enabled confinement of over 30 milliseconds.

Edge Biasing Electrodes are placed at the plasma edge to apply a radial electric field. This creates rotation due to E×B drift, improving plasma stability and confinement.

Advanced Fuel Focus TAE’s ultimate goal is proton-boron-11 reaction (p-$^{11}$B). This reaction produces almost no neutrons, greatly reducing radioactivation problems. However, since the reaction cross-section peaks at high temperature (about 600 keV), it involves more difficult technical challenges than current D-T fusion.

University of Wisconsin-Madison

The plasma physics research group at the University of Wisconsin is promoting modern research on axisymmetric mirror confinement.

WHAM (Wisconsin HTS Axisymmetric Mirror) A plan for an axisymmetric mirror experimental device using high-temperature superconducting (HTS) magnets. The aim is to achieve dramatic improvement in confinement performance by achieving much higher mirror ratios (R > 20) than conventional mirror devices.

Main research themes:

• Verification of collisional confinement scaling at high mirror ratios
• Ion distribution control by oblique neutral beam injection
• MHD stability limits in axisymmetric configurations

The WHAM project explores the feasibility of next-generation mirror devices utilizing high-temperature superconducting technology, building on the success of GDT.

Continued Research in Russia

At the Budker Institute of Nuclear Physics, “GDMT (Gas Dynamic Multiple-mirror Trap)” is being considered as an advanced version of GDT. By arranging multiple mirror cells in series, longer confinement times and higher fusion output are targeted.

Conceptual studies on fusion-fission hybrid reactors using GDT as a neutron source are also underway. The 14 MeV neutrons generated in the mirror device are used for fission reactions of fission fuel (thorium-232 or uranium-238) arranged around it. This hybrid approach can achieve energy multiplication without achieving fusion conditions (Q > 1) and is expected to be practical in the near future.

Developments in China

Research on mirror confinement is also being conducted in China. At the Institute of Plasma Physics, Chinese Academy of Sciences (ASIPP), conceptual studies on mirror-type neutron sources are underway. There is also high interest in fusion-fission hybrid reactors, and mirror-type driver sources are being considered.

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Characteristics and Advantages of the Mirror Approach

Inherent Advantages of Open-Ended Systems

The mirror approach is classified as “open-ended magnetic confinement.” Compared to toroidal devices (tokamaks, stellarators), it has the following inherent advantages.

Ease of Steady-State Operation In tokamaks, the central solenoid flux is consumed to induce plasma current. This inherently means pulsed operation, and steady-state operation requires complex current drive techniques.

In mirror devices, induced current is not required, and plasma is confined by external magnetic fields alone. Therefore, steady-state operation is possible in principle, facilitating operation as a power plant.

Simple Structure As a linear device, the complex three-dimensional magnetic field coil arrangement of toroidal devices is not required. This leads to reduced construction costs and ease of maintenance.

Particularly in blanket design (structures that receive neutrons and produce tritium in fusion reactors), cylindrical shapes are easy to manufacture and replace. Compared to the complex toroidal shape of tokamaks, there is an advantage in engineering feasibility.

Continuous Fuel and Ash Processing Plasma escaping from the open ends enables clear separation of fuel injection and exhaust points. Unreacted fuel recovery and recycling, and helium ash removal can be performed continuously.

In tokamaks, these processes are performed in the divertor region, but since it is on the same field lines as the confined plasma, impurity backflow becomes problematic. In mirror devices, the ends are magnetically isolated from the main plasma, so this problem is mitigated.

High-Beta Operation

The beta value $\beta$ is defined as the ratio of plasma pressure to magnetic field pressure:

$$\beta = \frac{2\mu_0 \langle p \rangle}{B^2}$$

where $\langle p \rangle$ is the average plasma pressure and $B$ is the magnetic flux density.

In mirror devices, confinement of high-beta plasma close to $\beta \sim 1$ is possible. This is a much higher value compared to tokamaks (typically $\beta \sim 5\%$) or stellarators ($\beta \sim 5-10\%$).

Advantages of high-beta operation:

• Higher plasma pressure (fusion output) can be achieved at the same magnetic field strength
• Magnet cost and technical difficulty can be reduced
• Favorable for high-temperature, high-density conditions required for advanced fuels (p-$^{11}$B, etc.)

At GDT, confinement with β exceeding 60% has actually been demonstrated, experimentally confirming the high-beta capability of the mirror approach.

Application to Direct Energy Conversion

Charged particles escaping from open ends can be converted to electricity through direct energy conversion.

Electrostatic Direct Converters The escaping plasma is passed through a decelerating electric field to convert ion kinetic energy to electrostatic energy. By collecting this at electrodes, electricity is obtained directly without going through a thermal cycle.

Theoretical conversion efficiency reaches over 90%, far exceeding thermal cycle power generation (efficiency 30-40%). However, technology demonstration at practical scale has not yet been performed.

Compatibility with Advanced Fuels In D-T reactions, 80% of the generated energy is released as neutrons (which have no charge), so the benefit of direct conversion is limited.

However, in D-$^3$He or p-$^{11}$B reactions, most of the generated energy is released as charged particles (protons, alpha particles). For these advanced fuels, the advantages of direct conversion are maximized.

Since mirror devices have open ends, application to direct conversion is natural. In future advanced fuel fusion reactors, mirror-type direct converters are a promising option.

Application to Fusion-Fission Hybrids

Mirror devices are promising as fusion driver sources for fusion-fission hybrid reactors.

In hybrid reactors, 14 MeV neutrons generated by fusion reactions irradiate surrounding fission fuel (uranium-238, thorium-232) to induce fission chain reactions. Since fission energy multiplies fusion energy, overall energy gain is obtained even without achieving Q > 1 from fusion alone.

Reasons why mirror devices are suitable for hybrid reactors:

• Cylindrical shape is advantageous for blanket design
• Steady-state operation is possible
• Easy to achieve high neutron flux density
• System works even at Q < 1

The Russian GDT group is energetically pursuing research in this direction.

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Summary of Formulas

The main formulas necessary for understanding mirror confinement are summarized below.

Basic Quantities

Cyclotron frequency

$$\omega_c = \frac{|q|B}{m}, \quad f_c = \frac{\omega_c}{2\pi}$$

Larmor radius

$$r_L = \frac{mv_\perp}{|q|B}$$

Magnetic moment (adiabatic invariant)

$$\mu = \frac{mv_\perp^2}{2B} = \text{const.}$$

Mirror force

$$F_\parallel = -\mu \nabla_\parallel B = -\mu \frac{\partial B}{\partial s}$$

Mirror Confinement

Mirror ratio

$$R_m = \frac{B_{\max}}{B_{\min}}$$

Confinement condition

$$\sin^2\theta = \frac{v_\perp^2}{v^2} > \frac{1}{R_m}$$

Critical pitch angle

$$\theta_c = \arcsin\left(\frac{1}{\sqrt{R_m}}\right)$$

Loss cone half-apex angle

$$\alpha = \arccos\left(\frac{1}{\sqrt{R_m}}\right), \quad \tan\alpha = \sqrt{R_m - 1}$$

Confinement Time

90-degree deflection time (ion-ion collision)

$$\tau_{ii} = \frac{12\pi^{3/2}\varepsilon_0^2 m_i^{1/2} T_i^{3/2}}{n_i q^4 \ln\Lambda}$$

Simple mirror confinement time

$$\tau_p \approx \tau_{ii} \cdot R_m \cdot \ln R_m$$

Tandem mirror confinement time (potential confinement)

$$\tau_p \approx \tau_{ii} \cdot R_m \cdot \left(\frac{e\phi_c}{T_i}\right) \cdot \exp\left(\frac{e\phi_c}{T_i}\right)$$

MHD Stability

Flute instability growth rate (approximate)

$$\gamma \sim \sqrt{\frac{2p}{R_c \rho}}$$

Stability integral (stable if negative)

$$\int \kappa \cdot \nabla p \, dl < 0$$

Beta Value

$$\beta = \frac{2\mu_0 \langle p \rangle}{B^2} = \frac{n(T_i + T_e)}{B^2/(2\mu_0)}$$

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Future Outlook

Integration with High-Temperature Superconducting Technology

Recent rapid advances in high-temperature superconducting (HTS) technology are opening new possibilities for mirror confinement. High-temperature superconducting magnets using REBCO (rare-earth barium copper oxide) tape have made magnetic fields above 20 T achievable, which were difficult with conventional low-temperature superconductors.

Benefits of higher magnetic fields:

• Improved mirror ratio ($R_m \propto B_{\max}/B_{\min}$)
• Narrower loss cone and improved confinement time
• Increased fusion output in high-beta plasmas

The WHAM project at the University of Wisconsin is a pioneering effort to apply this technology to axisymmetric mirrors.

Path to Compact Fusion Reactors

The simple structure and high-beta capability of the mirror approach are advantageous for realizing compact fusion reactors.

In tokamaks, many technical challenges exist, including plasma current drive, disruption countermeasures, and complex blanket structures. These lead to device enlargement and cost increases.

In the mirror approach, many of these challenges are avoided or mitigated. Combined with high-temperature superconductors, there is the possibility of realizing economically competitive compact reactors.

Application to Space Propulsion

Mirror confinement plasma is also being considered for application to space propulsion systems.

Plasma escaping from the ends through magnetic nozzles can be used as propellant. The exhaust velocity of high-temperature plasma heated by fusion reactions far exceeds that of chemical rockets.

For deep space missions such as crewed Mars exploration, propulsion systems that balance high specific impulse and high thrust are required. Fusion mirror propulsion is a promising candidate to meet this requirement.

Challenges for the Research Community

Mirror confinement research suffered significant damage to its knowledge base and human resources due to program reductions in the 1980s. Currently, many experts in this field are aging, and technology transfer is a challenge.

On the other hand, new vitality is being generated by the rise of private fusion companies and the entry of young researchers. Inheriting traditional knowledge to the next generation while incorporating modern technology (high-temperature superconductors, advanced diagnostics, computational science) is essential for the revival of mirror research.

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Related Topics

• Confinement Methods: Overview - Overall picture of fusion confinement methods
• Tokamak Approach - Representative of toroidal magnetic confinement
• Stellarator/Helical Approach - Confinement by external coils only
• Magnetic Configurations - Comparison of various magnetic configurations
• Plasma Physics: Particle Motion - Basic motion of charged particles
• Plasma Heating - Neutral beam injection, electron cyclotron heating
• MHD Instabilities - Explanation of flute instability, etc.

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References

The following literature is useful for comprehensive explanations of mirror confinement:

• Post, R.F., “The magnetic mirror approach to fusion,” Nuclear Fusion 27, 1579 (1987)
• Baldwin, D.E. and Logan, B.G., “Improved tandem mirror fusion reactor,” Physical Review Letters 43, 1318 (1979)
• Ryutov, D.D., “Open-ended traps,” Soviet Physics Uspekhi 31, 300 (1988)
• Miyoshi, S., “Potential Confinement in Tandem Mirrors,” Journal of Plasma and Fusion Research (2000)